====== Exhaustion ======

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\( \newcommand{\diff}[2]{\frac{d{#1}}{d{#2}}} \)
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So far we've looked at some special forms of mating processes, such as [[Estrous1GenderRedux | estrous]], [[Monogamy1Gender | monogamy]], and [[Gestation1Gender | gestation]].  Each of these cases exhibited roughly logistic dynamics for high population densities with mate-finding corrections for low densities that produced a weak Allee effect, impairing the population growth rate.  Is this result general?  How can we know?  In this page I will explore a more general class of sexual models and look for universal properties.

====== Asexual logistic kinetics ======

The logistic equation derives from three simple processes((actually, //spontaneous death// isn't required)):
\[
\begin{array}{rcll}
  N   & \xrightarrow{\alpha}   & 2 N       & \text{(asexual reproduction)} \\
  N   & \xrightarrow{\delta}   & \emptyset & \text{(spontaneous death)} \\
  2 N & \xrightarrow{\chi}     & N         & \text{(competition).}
\end{array}
\]

From these three processes we derive \(\diff{n}{t} = (\alpha - \delta) n - \chi n^2 = r n (1-n/K) \) where \(r = \alpha -\delta \) and \(K = (\alpha - \delta) /\chi \).  If \(\alpha > \delta \) then the population stabilizes at \(K > 0 \), the //carrying capacity//.  Notice that this derived from an asexual reproduction process where a single individual, \(N\), spontaneously produces two.

====== Naive sexual kinetics ======

What is the natural extension of this system to sexual species, where a mating event between two parents is required to produce an offspring?  The simplest extension to the logistic kinetics would be
\[
\begin{array}{rcll}
  N & \xrightarrow{\delta}  & \emptyset & \text{(spontaneous death)} \\
  2 N & \xrightarrow{\beta} & 3 N       & \text{(sexual reproduction)} \\
  2 N & \xrightarrow{\chi}  & N         & \text{(competition)}
\end{array}
\]
where the asexual reproduction process is replaced with a sexual process (requiring a partner).

But this can't be a correct description of what is found in nature because it produces unstable dynamics: \(\diff{n}{t} = (\beta - \chi) n^2 - \delta n \).  For a high birth rate (\( \beta > \chi \)) there is an unstable equilibrium at \(n^*=\delta/(\beta - \chi) \).  Large populations (\( n>n^* \)) will diverge and small populations (\( n<n^* \)) will collapse.  Populations with an insufficient birth rate (\( \beta < \chi \)) will also collapse.

This is clearly an unsatisfactory description of real populations.

====== Minimal ingredients ======

We know the logistic equation provides a reasonable description of many sexual populations.  But our naive sexual model proved unstable, and hence unrealistic.  Clearly, we require a more nuanced description of sexual population processes.  But how much do we need to add to get reasonable behaviour (ie. a stable equilibrium)?  What are the minimal ingredients that need to be added?  And when we find a satisfactory description, how will the dynamics compare to the ubiquitous logistic equation?

  * argue against higher than pairwise interactions
  * or allow higher order and show it can produce a strong Allee effect in the undifferentiated model ([[#Bewernick et al., 2005]])
  * differentiate into two types and then stick to pairwise (to avoid combinatorial explosion and because it will prove sufficient)
  * note: later on should argue that weak Allee effect is interesting.  Sure, pop still grows deterministically but stochastic effects become strong for small populations.  A weak Allee effect delays recovery and puts the population at increased risk of stochastic extinction.  (This argument isn't necessary to make the case for our extension though, because we find a **strong** Allee effect)

===== Undifferentiated individuals =====

As a first guess let's consider the following simple model: let \(N\) represent an individual.  We deliberately coarse grain all individuals so the only information available to us is their presence or absence.  No states of the individual are differentiated.  Let's require simple competition to occur within the population but also allow any other arbitrary process:
\[
\begin{array}{rclcl}
  a_i N & \xrightarrow{\alpha_i} & b_i N & & \text{(arbitrary reaction } i\textrm{)} \\
  2 N   & \xrightarrow{\delta}   & N     & & \text{(competition).}
\end{array}
\]

For this set of reactions the population dynamics will follow
\[ \diff{n}{t} = \sum_i \alpha_i (b_i-a_i) n^{a_i} - \delta n^2 \]
which can be written as
\[ \diff{n}{t} = \sum_j c_j n^j - \delta n^2 \equiv f(n) - \delta n^2 \]
where
\[ c_j \equiv \sum_i \delta_{a_i, j} \alpha_i (b_i - j) \]
so that \(f(n)\) is any function that can be represented by a series expansion.

In principle we can capture practically any dynamics we choose with this form.  For example, we've already encountered behaviour like \( f(n) = \beta n^2/(n+n_\mathrm{Allee}) \) demonstrating a weak Allee effect due to difficult mate finding at low densities.  In this case we can use the Taylor expansion to find
\[ f(n) = \beta \frac{n^2}{n+n_\mathrm{Allee}} = \sum_{j\geq 1} (-1)^{j+1} \frac{n^j}{n_\mathrm{Allee}^j}.\]
Hence, \( c_0=0, c_{j\geq 1}=(-1)^{j+1}/n_\mathrm{Allee}^j \) which can be constructed from suitably chosen reaction processes.

This example demonstrates a conceptual weakness of this approach: producing the desired \(f(n)\) may require an arbitrarily complicated set of interactions (reactions) between many individuals simultaneously.  What was intended as a simple reproduction of the Allee effect we had discovered elsewhere has turned out to require an extremely careful choice of an infinite set of reactions.  It is unlikely finely-tuned reaction-sets satisfying the necessary constraints should occur naturally.

====== Differentiated Types ======

Since high-order reactions (with many reactants) are expected to be exceedingly rare our model should not depend on them.  Low-order -- especially singleton (spontaneous) and pairwise -- reactions should largely account for the population dynamics.  A model that captures the dynamics with only singleton and pairwise reactions is more convincing than one that depends on higher order processes.

===== Constraints =====

====== Bookmark ======

  * 1 or 2 reactants
  * only change by one individual per reaction (arbitrary)

With these restrictions the asexual and sexual processes describe all the possibilities (neglecting immigration).  We can only incorporate more details by adding another level of complexity: differentiating individuals into types.  Let's break the population into two types: call them \(A\) and \(B\).  There are a few different ways we could do this but one fairly option is to differentiate by births--only individuals of type \(B\) are able to reproduce.  \(B\) is the //birthing// type.  We will discover restrictions on the possible interpretations of \(B\) (eg. gender, sexual maturity, estrous) that allow for biologically meaningful dynamics.


====== References ======

<BOOKMARK:bewernick_et_al_2005>(Bewernick et al., 2005) Robert L. Bewernick, Jeremy D. Dewary, Eunice Grayz, and Nancy Y. Rodriguez, "Population Processes", AMSSI Technical Report, http://www.public.asu.edu/~etcamach/AMSSI/page4.html (2005)